L(s) = 1 | − 2.73·3-s + 3.46·5-s − 2.73·7-s + 4.46·9-s + 1.26·11-s − 5.46·13-s − 9.46·15-s − 17-s + 1.46·19-s + 7.46·21-s − 1.26·23-s + 6.99·25-s − 3.99·27-s − 3.46·29-s + 4.19·31-s − 3.46·33-s − 9.46·35-s − 4.53·37-s + 14.9·39-s − 6·41-s + 8.39·43-s + 15.4·45-s − 6.92·47-s + 0.464·49-s + 2.73·51-s − 12.9·53-s + 4.39·55-s + ⋯ |
L(s) = 1 | − 1.57·3-s + 1.54·5-s − 1.03·7-s + 1.48·9-s + 0.382·11-s − 1.51·13-s − 2.44·15-s − 0.242·17-s + 0.335·19-s + 1.62·21-s − 0.264·23-s + 1.39·25-s − 0.769·27-s − 0.643·29-s + 0.753·31-s − 0.603·33-s − 1.59·35-s − 0.745·37-s + 2.39·39-s − 0.937·41-s + 1.27·43-s + 2.30·45-s − 1.01·47-s + 0.0663·49-s + 0.382·51-s − 1.77·53-s + 0.592·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 3 | \( 1 + 2.73T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 4.53T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8.39T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 - 0.535T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 - 2.53T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.864522587100450236358327443498, −9.039212865467245279814440693438, −7.39132897354886857711951238396, −6.56909116194763809374856608461, −6.09622975587532286860639395999, −5.33614368183299603613454959117, −4.58295074127397763497136979886, −2.93683579973442792028204234987, −1.64459928959245029252102286296, 0,
1.64459928959245029252102286296, 2.93683579973442792028204234987, 4.58295074127397763497136979886, 5.33614368183299603613454959117, 6.09622975587532286860639395999, 6.56909116194763809374856608461, 7.39132897354886857711951238396, 9.039212865467245279814440693438, 9.864522587100450236358327443498